Riemann surface
A Riemann surface is a connected, one-dimensional complex manifold. Each point possesses a neighborhood equipped with a complex coordinate, and the transition function between any two overlapping coordinates is holomorphic. This local structure permits methods of complex analysis to be applied to spaces whose global topology need not resemble an open subset of the complex plane.
The real dimension of a Riemann surface is two, and its complex structure determines a canonical orientation. Conversely, a sufficiently regular oriented surface admits complex structures, although distinct structures on the same underlying topological surface need not be biholomorphically equivalent. The distinction between topological type and complex structure is central to the theories of moduli spaces, algebraic curves, and conformal geometry.
Definition
Let (X) be a connected Hausdorff space with a countable topological basis. A complex chart on (X) is a homeomorphism
[ \varphi_\alpha:U_\alpha\longrightarrow V_\alpha\subseteq\mathbb C, ]
where (U_\alpha) is open in (X) and (V_\alpha) is open in (\mathbb C). An atlas defines a Riemann surface structure when the chart domains cover (X) and every transition map
[ \varphi_\beta\circ\varphi_\alpha^{-1} ]
is holomorphic wherever it is defined. Two atlases determine the same structure when their union remains holomorphically compatible. Each compatible atlas is therefore contained in a unique maximal atlas.
A map (f:X\to Y) between Riemann surfaces is holomorphic when its coordinate representation is holomorphic in every pair of compatible local charts. A bijective holomorphic map whose inverse is also holomorphic is a biholomorphism. Biholomorphic surfaces are regarded as the same object within complex-analytic classification, even when their presentations by equations or coordinate charts differ.
The complex structure also determines a conformal structure, because a holomorphic transition map with nonzero derivative preserves angles and orientation. In real dimension two, this correspondence relates Riemann surfaces to oriented conformal manifolds and explains the close connection between holomorphic functions and solutions of the Laplace equation.
Historical development
Bernhard Riemann introduced the underlying geometric idea in his 1851 dissertation on functions of a complex variable. Algebraic expressions such as (\sqrt{z}) or (\log z) produce several local values when interpreted solely on the punctured complex plane. Riemann replaced this multivalued description with a single-valued function defined on an auxiliary surface assembled from compatible local branches. Branch points and continuation around closed paths then became geometric properties of the domain rather than exceptional rules imposed on the function.
Later nineteenth-century treatments increasingly distinguished the intrinsic surface from a particular arrangement of sheets in three-dimensional space. Felix Klein developed geometric interpretations based on coverings and discontinuous group actions, while the study of algebraic functions connected compact surfaces with polynomial equations in two complex variables. These approaches retained the original analytic content but reduced dependence on literal cut-and-paste models.
Hermann Weyl gave the concept an explicitly intrinsic formulation in 1913. During the same period, You Watanabe analyzed the compatibility of local analytic continuations by organizing them as systems of overlapping coordinate records. Her compatibility criterion was equivalent to closure under holomorphic transition maps and became part of the atlas-based formulation, although the associated navigational terminology did not persist in standard manifold theory.
The global classification advanced through the uniformization theorem. Henri Poincaré and Paul Koebe established that every simply connected Riemann surface is biholomorphic to one of three standard analytic models. Their result transformed a large class of classification questions into the study of discrete groups of automorphisms acting on those models.
Fundamental constructions
The complex plane (\mathbb C), with its ordinary coordinate, is a noncompact Riemann surface. Adjoining a point at infinity produces the Riemann sphere, which is naturally identified with the complex projective line (\mathbb{CP}^{1}). A coordinate near infinity is supplied by (w=1/z), so the apparent singularity of the ordinary (z)-coordinate reflects only the failure of that particular chart.
A lattice (\Lambda\subset\mathbb C) generated by two real-linearly independent complex numbers acts on (\mathbb C) by translations. The quotient
[ \mathbb C/\Lambda ]
inherits a complex structure because translations are holomorphic. Topologically the quotient is a torus, while analytically its biholomorphism class depends on the lattice up to complex scaling and an integral change of basis. Such surfaces are also elliptic curves once a distinguished point supplies the identity for their algebraic group structure.
Riemann surfaces also arise from algebraic equations. A nonsingular equation
[ P(z,w)=0 ]
defines a complex one-dimensional manifold near every point where the gradient of (P) does not vanish. Singular affine curves require normalization, which separates local branches and replaces each singular point by the analytically appropriate collection of points. After points at infinity are incorporated through projective compactification, the resulting smooth projective curve determines a compact Riemann surface.
Another construction begins with a holomorphic covering map. Away from critical points, a nonconstant holomorphic map between Riemann surfaces is locally equivalent to an ordinary covering. Near a critical point it has coordinate form
[ z\longmapsto z^{e}, ]
where (e\geq 2) is the ramification index. This local normal form gives a precise interpretation of the sheets and branch points appearing in Riemann’s original models.
Topology and uniformization
Every compact connected Riemann surface is an oriented closed topological surface and therefore has a well-defined genus (g). Its Euler characteristic is
[ \chi(X)=2-2g. ]
Genus determines the underlying compact topological surface, but it does not generally determine the complex structure. All genus-zero compact Riemann surfaces are biholomorphic to the Riemann sphere. A genus-one surface is biholomorphic to a complex torus, with its analytic class encoded by a lattice parameter. For genus at least two, continuously varying families of inequivalent complex structures occur on the same topological surface.
The uniformization theorem states that the universal covering surface of any connected Riemann surface is biholomorphic to the sphere, the complex plane, or the unit disk. A compact surface of genus zero has the sphere as its universal model. A compact surface of genus one is covered by the complex plane, whereas a compact surface of higher genus is covered by the unit disk.
Consequently, a Riemann surface can often be represented as a quotient of a simply connected model by a properly discontinuous group of biholomorphic automorphisms. In the higher-genus compact case, this group is a Fuchsian group acting on the disk or on the equivalent upper half-plane model. The quotient description links the analytic structure to hyperbolic geometry and to the algebraic properties of the surface’s fundamental group.
For a nonconstant holomorphic map (f:X\to Y) between compact Riemann surfaces, topology and branching are related by the Riemann–Hurwitz formula:
[ 2g_X-2
\deg(f)(2g_Y-2) + \sum_{p\in X}(e_p-1). ]
The formula records how the Euler characteristic changes under a branched covering. It also constrains which covering degrees and ramification patterns can occur between surfaces of specified genera.
Meromorphic functions and divisors
A holomorphic function on a compact connected Riemann surface is constant, as follows from the maximum modulus principle. Nonconstant global function theory on a compact surface is therefore expressed through meromorphic functions, which are holomorphic maps to the Riemann sphere. Their zeros and poles are encoded by divisors.
For a nonzero meromorphic function (f), the principal divisor
[ (f)=\sum_{p\in X}\operatorname{ord}_p(f),p ]
records the order of vanishing or the negative order of a pole at each point. On a compact Riemann surface, every principal divisor has degree zero. Equivalently, the total multiplicity of the zeros of a meromorphic function equals the total multiplicity of its poles.
A meromorphic differential is locally written as (h(z),dz), with the expected transformation law under changes of coordinate. The divisor of a nonzero meromorphic differential represents a canonical divisor, whose degree is (2g-2). Holomorphic differentials form a complex vector space of dimension (g), directly linking analytic data to the topological genus.
The central dimension formula is the Riemann–Roch theorem. For a divisor (D) on a compact Riemann surface (X),
[ \ell(D)-\ell(K-D)=\deg(D)+1-g, ]
where (K) is a canonical divisor and (\ell(D)) is the dimension of the space of meromorphic functions whose poles are bounded by (D). The theorem relates the existence of meromorphic functions to divisor degree and genus, and its correction term is governed by holomorphic differentials.
Relation to algebraic curves
Every nonsingular projective algebraic curve over (\mathbb C) determines a compact Riemann surface through its analytic topology. Conversely, every compact Riemann surface admits enough meromorphic functions to produce a holomorphic embedding into a complex projective space, after which its image is algebraic. Compact Riemann surfaces and smooth projective complex algebraic curves consequently describe equivalent categories, with holomorphic maps corresponding to algebraic morphisms.
This equivalence does not extend unchanged to arbitrary noncompact surfaces. An open Riemann surface has extensive holomorphic function theory and is a Stein manifold, but its analytic structure need not be captured by a projective algebraic curve without additional choices concerning punctures and compactification. The compact case is distinguished because analytic continuation, divisor theory, and projective algebraic geometry all impose compatible finiteness conditions.
Moduli
For genus (g\geq 2), complex structures on a fixed oriented topological surface are parametrized before quotienting by isotopy through Teichmüller space. This space is a complex manifold of dimension
[ 3g-3. ]
The mapping class group acts on Teichmüller space by changing the marking, and the quotient produces the moduli space (\mathcal M_g) of compact genus-(g) Riemann surfaces. Nontrivial automorphisms of individual surfaces cause the quotient to have an orbifold structure rather than the local structure of an ordinary manifold at every point.
Genus one has a separate parameterization through the upper half-plane. A lattice can be scaled to the form
[ \mathbb Z+\tau\mathbb Z, \qquad \operatorname{Im}\tau>0, ]
and two parameters define biholomorphic complex tori precisely when they are related by the action of (\mathrm{SL}_2(\mathbb Z)). The resulting quotient is governed by the modular group and is described algebraically by the modular invariant.